fluctuation force laboratory
The Gap Is Made of Both Sides
Move ideal mirrors, then replace them with dispersive materials and an intervening medium. A live Lifshitz sum exposes attraction, repulsion, spectral sign changes, and a force-zero separation without treating vacuum energy as a measured substance.
Layer 1, the ideal lawBring two perfect mirrors closer. The pressure does not merely double. Halving the gap multiplies its magnitude by sixteen.
signed pressure
P = -pi2 hbar c / (240 a4)
force on chosen area
F = P times A
same force as a weight
m = |F| / standard gravity
The expression above has a narrow jurisdiction: infinite, perfectly conducting, parallel plates, zero temperature, no roughness, and no edge. Its sign matters. At a = 100.0 nm the live engine gives ; at 1.000 micrometre it gives . Those values use the exact SI definitions of h and c, with hbar = h/(2 pi).
The cutoff is allowed to disappear, the divergent pieces are not observables
Casimir's 1948 route introduced a high-frequency cutoff and used Euler-Maclaurin analysis. It did not use the Riemann zeta function. The panel below regulates both the discrete mode sum and its continuum counterpart with the same exponential, then subtracts them. Each side grows as the cutoff is removed. Their separation-dependent difference tends to -1/360, which supplies the finite plate energy.
2,000-mode sum
continuum integral
difference / (-1/360)
This is an exponential regulator demonstration of the ideal mode calculation, not a material transparency model. The sum is truncated after 2,000 terms. At the smallest available lambda the omitted exponential tail is negligible in double precision, while cancellation between two large numbers limits the trustworthy digits.
The mirrors were doing more work than the vacuum
A perfect boundary is a limit, not a sample. Jaffe showed that the same force may be formulated as a relativistic quantum interaction among charges and currents. In that description it vanishes as the electromagnetic coupling tends to zero, while the ideal-conductor result is the opposite limiting case. This removes any need to treat absolute zero-point energy as a literal mechanical substance. It does not make the effect classical.
The observable is the boundary-dependent interaction energy or stress. Quantized-field modes, fluctuational electrodynamics, and quantum interactions among matter agree on that observable. A force measurement does not choose one of those equivalent bookkeeping languages as an ontology.
Layer 2, the material answerThe gap has a spectrum
The sophisticated dismissal is correct as far as it goes: the ideal law is a toy. The further result is not a correction percentage. It is a different calculation, built from each material's causal dielectric response at imaginary frequency. Every Matsubara band can pull or push, and their balance changes with separation.
These are constructed oscillator models, not fits to named substances. Their coefficients are printed below. The point is to operate the equation and its sign logic without laundering a toy spectrum into material data.
repulsive pressureattractive pressureselected gap and force zero
signed pressure at selected gap
zero found on 50 to 1,000 nm scan
Open the sum one band at a time
selected band contribution
numerical tail diagnostic
absolute final retained term / absolute total, not a rigorous tail bound
For nonmagnetic dielectrics, the zero-frequency TE term in this engine is zero. The TM term is retained with half weight, as the prime on the sum requires. For the optional Drude metal, epsilon(0) is treated as infinite and the TE zero term is also set to zero. That convention is a declared model choice, not a settlement of the metal thermal question.
The measured repulsion this model does not impersonate
Munday, Capasso, and Parsegian used a gold sphere and a silica plate immersed in bromobenzene. The plotted span is approximately 20 to 300 nm; the displayed curve averages 50 data sets; and the authors state that magnitudes below 10 pN cannot be determined accurately. This was a solid-liquid-solid repulsion, not two plates repelling in vacuum. The paper attributes significant theory-data discrepancy probably to optical-property uncertainty and says its two-oscillator bromobenzene model is insufficient for detailed analysis.
Svetovoy and colleagues measured separately prepared gold films over 0.14 to 33 micrometres. The extracted plasma energies ranged from 6.8 to 8.4 eV. Lifshitz forces computed from those responses were 5 to 14 percent smaller at 100 nm than a customary handbook-response prediction. That is a calculation from measured optics, not a directly measured force residual, and it is not a comparison with perfect conductors.
Numbers with their dependencies still attached
recomputing the shipped engine
| quantity | live result | how it can fail | status |
|---|
Constants fixed. The engine uses the exact SI values h = 6.62607015e-34 J s, c = 299792458 m/s, k_B = 1.380649e-23 J/K, and eV = 1.602176634e-19 J. It derives hbar = h/(2 pi). Weight-equivalent uses conventional standard gravity g0 = 9.80665 m/s2; it is an analogy, not a local mass measurement.
Free choices fixed. Every displayed Lifshitz pressure is for plane parallel half-spaces. The default temperature is 293.15 K. Dielectric functions are the visible oscillator lists, whose strengths and resonance energies are choices, not recovered sample data. The calculation retains Matsubara indices n = 0 through 160, applies half weight at zero, uses generated 48-point Gauss-Legendre quadrature over y_n to y_n + 36, samples 41 logarithmically spaced separations from 50 to 1,000 nm, and refines a bracketed zero with 20 bisections. The ideal cutoff view uses 2,000 terms.
Uncertainties named. The oscillator presets omit real absorption structure, anisotropy, magnetic response, surface layers, spatial dispersion, roughness, patch potentials, finite thickness, edge effects, and non-planar geometry. The Matsubara tail is truncated, the transformed integral is cut off, and quadrature uses double precision. The last-term ratio is only a diagnostic, not a bound. Root classification uses the sign on the two sides of the refined bracket. A sign-changing pressure in this plane-plane model does not by itself prove levitation in a finite experimental geometry. Real predictions require Kramers-Kronig-consistent optical data and an explicit low-frequency extrapolation.
What is transcribed rather than recomputed. The Munday measurement counts and range, and the Svetovoy optical ranges and percentages, are source records. Browser arithmetic cannot validate a paper transcription. The offline verifier checks that the page has not drifted from a separately entered source record, while the reader must still open the papers to audit that record.
Run node research/casimir-lifshitz/verify-casimir-lifshitz.mjs. It independently recomputes the equations, executes this page's own engine in a VM, and deliberately mutates constants and logic to prove the checks can turn red.
The zero mode is still an argument
As of 2026-08-01, the finite-temperature prediction for real metals still depends sharply on how the transverse-electric zero-frequency response is extrapolated. A dissipative Drude model omits that TE zero contribution; a lossless plasma extrapolation retains one. The experimental and thermodynamic interpretation remains disputed. This page chooses Drude when its metal model is selected, prints that choice beside the calculation, and does not average the alternatives into a false consensus.
The nearby engineering edge is more concrete: material-specific force prediction is limited by the optical response of the actual deposited films, especially unmeasured low frequencies, and by electrostatic and geometric backgrounds. The gold-film study's calculated spread at 100 nm is already larger than many tidy textbook corrections. Better force data cannot repair dielectric data from a different sample.